Graph representation learning received increasing attentions in recent years. Most of the existing methods ignore the complexity of the graph structures and restrict graphs in a single constant-curvature representation space, which is only suitable to particular kinds of graph structure indeed. Additionally, these methods follow the supervised or semi-supervised learning paradigm, and thereby notably limit their deployment on the unlabeled graphs in real applications. To address these aforementioned limitations, we take the first attempt to study the self-supervised graph representation learning in the mixed-curvature spaces. In this paper, we present a novel Self-Supervised Mixed-Curvature Graph Neural Network (SelfMGNN). To capture the complex graph structures, we construct a mixed-curvature space via the Cartesian product of multiple Riemannian component spaces, and design hierarchical attention mechanisms for learning and fusing graph representations across these component spaces. To enable the self-supervised learning, we propose a novel dual contrastive approach. The constructed mixed-curvature space actually provides multiple Riemannian views for the contrastive learning. We introduce a Riemannian projector to reveal these views, and utilize a well-designed Riemannian discriminator for the single-view and cross-view contrastive learning within and across the Riemannian views. Finally, extensive experiments show that SelfMGNN captures the complex graph structures and outperforms state-of-the-art baselines.